Video Transcript
Find the solution set of two 𝑥 minus one is less than or equal to negative nine given that 𝑥 is a natural number.
The natural numbers are all the positive integers or counting numbers: one, two, three, and so on. We need to solve the inequality and then identify which natural numbers satisfy this. In order to solve the inequality, we will use the balancing method and our knowledge of inverse operations.
The opposite or inverse of subtracting one is adding one. So we do this to both sides of the inequality. Negative nine plus one is equal to negative eight. Therefore, two 𝑥 is less than or equal to negative eight. Our next step is to divide both sides by two. Two 𝑥 divided by two is 𝑥, and negative eight divided by two is negative four. The solution to the inequality two 𝑥 minus one is less than or equal to negative nine is 𝑥 is less than or equal to negative four. This means that 𝑥 could be any value that is less than or equal to negative four.
However, in this case, we were only looking for positive integers. As there are no positive integers less than or equal to negative four, there are no natural number solutions to the inequality. We denote the fact there are no solutions as shown. This is sometimes known as the empty set or null set. It is a solution set with no values.
